HEAVISIDE FUNCTION IN PRECALCULUS (WITH SOLVED PROBLEMS)

Introduction to the Heaviside Function

• Invented by Oliver Heaviside, a self-taught mathematician, physicist, and electrical engineer.

• Denoted typically by a capital letter H.

• Small letter h refers to a general function, not the Heaviside function.

Characteristics of the Heaviside Function

• Returns only two values: 0 or 1.

  • If the input value (x) is less than zero: returns 0.

  • If the input value (x) is equal to zero: returns 0.

  • If the input value (x) is greater than zero: returns 1.

• Can be applied in various fields, including:

  • Laplace transformation.

  • Electrical devices (used to define on/off states).

  • Medical devices for monitoring body functions.

Examples of the Heaviside Function

• H(23) returns 1 (23 > 0).

• H(0) returns 0.

• H(-15.2) returns 0.

• H(7.38) returns 1.

• H(1.35) returns 1.

• H(1) returns 1.

• H(-1) returns 0.

• H(-5) returns 0.

Important Notes

• Remember to denote the Heaviside function properly; using a small h is incorrect.

• In exam scenarios, problems may involve substituting x for specific values to find answers based on the Heaviside function.

Example Problems

• For an exam question:

  • Replace x with 7: H(7) returns 1, with an additional answer to retrieve, such as 5/2.

• Combine Heaviside function with other functions:

  • Signum() function and Identity function may be presented in problems.

  • Example: Signum(-25) = -1, H(3) = 1, Identity(-7) = -7. In the calculation: 9 - 1 - 1 = -2, and -7 + 9 = 2; thus the answer is -1.

Conclusion

• The identity function is now introduced.

• Further learning can be pursued through additional video resources.